Method for imaging subterranean formations

ABSTRACT

A method of imaging a subterranean formation traversed by a wellbore using a tool comprising a transmitter for transmitting electromagnetic signals through the formation and a receiver for detecting response signals. The tool is brought to a first position inside the wellbore, and the transmitter is energized to propagate an electromagnetic signal into the formation. A response signal that has propagated through the formation is detected. A derived quantity for the formation based on the detected response signal for the formation is calculated, and plotted against time. The tool may be moved to at least one other position within the wellbore where the procedure may be repeated to create an image of the formation within the subterranean formation based on the plots.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is a continuation-in-part application of U.S.application Ser. No. 10/897,585, filed on Jul. 23, 2004, which is acontinuation-in-part application of U.S. application Ser. No. 10/701,735filed on Nov. 5, 2003, each of which are incorporated herein byreference. In addition, applicants claim priority based on U.S.provisional application Ser. No. 60/639,941, filed on Dec. 29, 2004.

FIELD OF THE INVENTION

In one aspect the present invention relates to a method for imaging asubterranean formation traversed by a wellbore. In another aspect, thepresent invention relates to a method for determining at least first andsecond distances from a device to at least a first layer and a secondlayer in a formation.

In an embodiment, the invention relates to a method for locating ananomaly and in particular to finding the location of a resistive orconductive anomaly in a formation surrounding a borehole in drillingapplications. In another aspect the invention relates to a method topermit rapid identification and imaging of a formation anomaly.

BACKGROUND OF THE INVENTION

In logging while drilling (LWD) geo-steering applications, it isadvantageous to detect the presence of a formation anomaly ahead of oraround a bit or bottom hole assembly. There are many instances where“Look-Ahead” capability is desired in LWD logging environments.Look-ahead logging is to detect an anomaly at a distance ahead of thedrill bit. Some look-ahead examples include predicting an over-pressuredzone in advance, or detecting a fault in front of the drill bit inhorizontal wells, or profiling a massive salt structure ahead of thedrill bit. While currently available techniques are capable of detectingthe presence of an anomaly, they are not capable of determining thelocation of the anomaly with sufficient depth or speed, they are notcapable of detecting an anomaly at a sufficient distance ahead of a bitor bottom hole assembly.

In formation evaluation, the depth of investigation of most loggingtools, wire line or LWD has been limited to a few feet from theborehole. One such tool is disclosed in U.S. Pat. No. 5,678,643 toRobbins, et al. U.S. Pat. No. 5,678,643 to Robbins, et al. discloses anLWD tool for locating an anomaly. The tool transmits acoustic signalsinto a wellbore and receives returning acoustic signals includingreflections and refractions. Receivers detect the returning acousticsignals and the time between transmission and receipt can be measured.Distances and directions to detected anomalies are determined by amicroprocessor that processes the time delay information from thereceivers. As set forth above, the depth of investigation facilitated bythe tool is limited.

Another technique that provides limited depth of investigation isdisclosed in U.S. Pat. No. 6,181,138 to Hagiwara. This technique forlocating an anomaly utilizes tilted coil induction tools and frequencydomain excitation techniques. In order to achieve a depth ofinvestigation with such a tool, a longer tool size would be required.However, longer tools generally result in poor spatial resolution.

In order to increase depth capabilities, transient electromagnetic (EM)methods have been proposed. One such method for increasing the depth ofinvestigation is proposed in U.S. Pat. No. 5,955,884 to Payton, et al.The tool disclosed in this patent utilizes electric and electromagnetictransmitters to apply electromagnetic energy to a formation at selectedfrequencies and waveforms that maximize radial depth of penetration intothe target formation. In this transient EM method, the current isgenerally terminated at a transmitter antenna and temporal change ofvoltage induced in a receiver antenna is monitored. This technique hasallowed detection of an anomaly at distances as deep as ten to onehundred meters. However, while Payton discloses a transient EM methodenabling detection of an anomaly, it does not provide a technique fordetecting anomalies ahead of a drill bit.

Other references, such as PCT application WO/03/019237 also disclose theuse of directional resistivity measurements in logging applications.This reference uses the measurements for generating an image of an earthformation after measuring the acoustic velocity of the formation andcombining the results. This reference does not disclose a specificmethod for determining distance and direction to an anomaly.

When logging measurements are used for well placement, detection oridentification of anomalies can be critical. Such anomalies may includefor example, a fault, a bypassed reservoir, a salt dome, or an adjacentbed or oil-water contact. It would be beneficial to determine both thedistance and the direction of the anomaly from the drilling site.

Tri-axial induction logging devices, including wire-line and LWD devicesare capable of providing directional resistivity measurements. However,no method has been proposed for utilizing these directional resistivitymeasurements to identify the direction to an anomaly.

Moreover, there is no rapid method for rapidly presenting the distanceinformation in a discernible form to permit a driller to accuratelysteer a LWD BHA to a desired location. Present methods typically utilizeinversion modeling to estimate distances to formation features. Thisinversion process is one in which data is used to build a model of theformation that is consistent with the data. The time and computingresources required to perform inversion can be considerable, which mayresult in a delay of identification of formation features, such asreservoirs.

Accordingly, a new solution is needed for determining the distance froma tool to an anomaly. Particularly such a solution is needed for lookingahead of a drill bit. Furthermore, a real time solution having anincreased depth of analysis is needed so that the measurements can beimmediately useful to equipment operators. Lastly, a means of rapidlyidentifying or imaging the formation features or boundaries is neededfor geosteering applications

SUMMARY OF THE INVENTION

In a first aspect of the invention, there is provided a method forimaging a subterranean formation traversed by a wellbore. The method maybe implemented using a tool comprising a transmitter for transmittingelectromagnetic signals through the formation and a receiver fordetecting response signals. The method comprises steps wherein

the tool is brought to a first position inside the wellbore;

the transmitter is energized to propagate an electromagnetic signal intothe formation;

a response signal that has propagated through the formation is detected;

a derived quantity is calculated for the formation based on the detectedresponse signal for the formation;

the derived quantity for the formation is plotted against time.

Then the tool is moved to at least one other position within thewellbore, whereafter the steps set out above are repeated. Optionally,this can be done again. Then an image of the formation within thesubterranean formation is created based on the plots of the derivedquantity.

Optionally tool is then again moved to at least one more other positionwithin the wellbore and the whole procedure can be repeated again.

In another aspect of the invention there is provided a method there isprovided a method for determining a distance to an anomaly in aformation relative to a device in a wellbore. The method may beimplemented using a device including at least one transmitter and atleast one receiver. The method includes calculating at least one of anapparent conductivity and an apparent resistivity based on a detectedresponse, which may include a voltage response. The at least one ofapparent conductivity and apparent resistivity is monitored over time,and the distance to the anomaly is determined based on an observedchange of the one of apparent conductivity and apparent resistivity.

The formation may comprise at least three layers whereby the tool islocated in one of these layers. The distances from the tool to at leastthe other two layers can be determined using the method of theinvention.

In this method, at least first and second distances from a device to atleast a first and a second layer in a formation can be determined,whereby at least one of the first and second layers comprises aresistivity or conductivity anomaly.

The first and second distances to the anomaly may be determined based onobserved multiple changes of the one of apparent conductivity andapparent resistivity, which may each include a deflection point.

In a particular embodiment of the invention, a voltage response ismeasured over time, and the response is utilized to calculate theapparent conductivity or the apparent resistivity over a selected timespan. A time at which the apparent conductivity deviates from a constantvalue is determined, and this time may by utilized to ascertain thefirst distance that can correspond to the distance that the anomaly isaway from the device in the wellbore.

A subsequent deviation at a later time can be utilized to ascertain thesecond distance with may correspond to the distance that the secondlayer is away from the device in the wellbore. The difference betweenthe first and second distances may be indicative of the thickness of thefirst layer which may contain an anomaly.

In a particular embodiment of the invention, the distance to theanomaly, and the thickness thereof, is determined when the at least oneof apparent conductivity and apparent resistivity reaches an asymptoticvalue.

The device may comprise a logging tool and/or it may be provided in ameasurement-while-drilling section or a logging-while-drilling sectionof a drill string trailing a drill bit.

In another aspect, the monitored one of apparent conductivity andapparent resistivity is plotted against time, after which and device issubsequently moved to another position in the wellbore. Anelectromagnetic signal is then again transmitted using the transmitter;

one of apparent conductivity and apparent resistivity based on areceiver-detected response is again calculated;

the one of apparent conductivity and apparent resistivity is againmonitored over time; and

again plotted against time.

Thereafter, an image of the formation within the subterranean formationcan be created based on the plots.

The again monitored one of apparent conductivity and apparentresistivity can be plotted in the same plot as the plot wherein the oneof apparent conductivity and apparent resistivity was originallyplotted. The creating of the image of the formation can includeidentifying two or more inflection points on each plotted calculated oneof apparent conductivity and apparent resistivity curve and fitting acurve to the two or more inflection points.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention is described in detail below with reference to theattached drawing figures, wherein:

FIG. 1 is a block diagram showing a system in accordance with embodimentof the invention;

FIG. 2 is a flow chart illustrating a method in accordance with anembodiment of the invention;

FIG. 3 is a graph illustrating directional angles between toolcoordinates and anomaly coordinates;

FIG. 4A is a graph showing a resistivity anomaly in a tool coordinatesystem;

FIG. 4B is a graph showing a resistivity anomaly in an anomalycoordinate system;

FIG. 5 is a graph illustrating tool rotation within a borehole;

FIG. 6 is a graph showing directional components;

FIG. 7 is a graph showing the voltage response from coaxial V_(zz)(t),coplanar V_(xx)(t), and the cross-component V_(zx)(t) measurements forL=1 m, for θ=30°, and at salt distance D=10 m;

FIG. 8 is a graph showing the voltage response from coaxial V_(zz)(t),coplanar V_(xx)(t), and the cross-component V_(zx)(t) measurements forL=1 m, for θ=30°, and at salt distance D=100 m;

FIG. 9 is a graph showing apparent dip (θ_(app)(t)) for an arrangementas in FIG. 7;

FIG. 10 is a graph showing apparent conductivity (σ_(app)(t)) calculatedfrom both the coaxial (V_(zz)(t)) and the coplanar (V_(xx)(t)) responsesfor the same conditions as in FIG. 9;

FIG. 11 is a graph showing the ratioσ_(app-coplanar)(t)/σ_(app-coaxial)(t) for the same approach angle (θ)and salt face distance (D) as in FIG. 3;

FIG. 12 is a graph showing apparent dip θ_(app)(t) for the L=1 m toolassembly when the salt face is D=10 m away, for various angles betweenthe tool axis and the target;

FIG. 13 is a graph similar to FIG. 12 whereby the salt face is D=50 maway from the tool;

FIG. 14 is a graph similar to FIG. 12 whereby the salt face is D=100 maway from the tool;

FIG. 15 is a schematic showing apparent conductivity with a coaxialtool;

FIG. 16 is a graph showing voltage response of the coaxial tool of FIG.15 in a homogeneous formation for various formation resistivities;

FIG. 17 is a graph showing voltage response in a homogeneous formationas a function of formation resistivity at different times (t) for thesame coaxial tool;

FIG. 18 is a graph showing voltage response in a homogeneous formationas a function of formation resistivity for a larger transmitter-receiverspacing than in FIG. 17;

FIG. 19 is a schematic showing apparent conductivity with a coplanartool;

FIG. 20 graphically shows voltage response of the coplanar tool of FIG.19 with a transmitter/receiver separation of L=1 m, in a homogeneousformation as a function of formation resistivity at different times (t);

FIG. 21 graphically shows voltage response in a homogeneous formation asa function of formation resistivity for a larger transmitter-receiverspacing than in FIG. 20;

FIG. 22 is a graph showing voltage response as a function of t as givenby the coaxial tool of FIG. 15 in a two-layer formation at differentdistances from the bed;

FIG. 23 is a graph showing the voltage response data of FIG. 22 in termsof the apparent conductivity (σ_(app)(t));

FIG. 24 is similar to FIG. 23 except that the resistivities of layers 1and 2 have been interchanged;

FIG. 25 presents a graph comparing σ_(app)(t) of FIG. 23 and FIG. 24relating to d=1 m;

FIG. 26 shows a graph of σ_(app)(t) for various transmitter/receiverspacings L in case d=1 m;

FIG. 27 shows σ_(app)(t) plots for d=1 m and L=01 m for two resistivityratios;

FIG. 28 shows a graph of the σ_(app)(t) for the case d=1 m and L=01 m,for various resistivity ratios while the target resistivity is fixed atR₂=1 ohm-m;

FIG. 29 shows a comparison of apparent conductivity at large values of tfor coaxial responses where d=01 m and L=01 m as a function ofconductivity of the target layer while the local conductivity is fixedat 1 S/m;

FIG. 30 graphically shows the same data as FIG. 29 plotted as the ratioof target conductivity over local layer conductivity versus ratio of thelate time apparent conductivity over local layer conductivity;

FIG. 31 shows a graph containing apparent conductivity (σ_(app)(t))versus time for various combinations of d and L;

FIG. 32 graphically shows the relationship between ray-path andtransition time tc;

FIG. 33 shows a plot of the apparent conductivity (σapp(z; t)) in bothz- and t-coordinates for various distances d;

FIG. 34 is a schematic showing apparent conductivity with a coaxialtool;

FIG. 35 is a graph showing voltage response as a function of t as givenby the coaxial tool of FIG. 34 at different distances from the bed;

FIG. 36 is a graph showing the voltage response data of FIG. 35 in termsof the apparent conductivity (σ_(app)(t));

FIG. 37 is similar to FIG. 36 except that the resistivities of layers 1and 2 have been interchanged;

FIG. 38 presents a graph comparing σ_(app)(t) of FIG. 36 and FIG. 37relating to d=1 m;

FIG. 39 presents a graph of the same data as displayed in FIG. 36 butnow on a linear scale of apparent conductivity;

FIG. 40 shows a graph of σ_(app)(t) on a linear scale for varioustransmitter/receiver spacings L in case d=1 m;

FIG. 41 shows a graph of late time conductivity as a function of d forvarious transmitter/receiver spacings L;

FIG. 42 shows the σ_(app)(t) plots for d=5 m and L=01 m for variousresistivity ratios;

FIG. 43 shows a graph of the σ_(app)(t) for the case d=5 m and L=01 m,but for different resistivity ratios while the target resistivity isfixed at R₂=1 ohm-m;

FIG. 44 graphically shows a comparison of the late time apparentconductivity at t=1 second with a model calculation, for the case targetresistivity R₂=1 ohm-m;

FIG. 45 graphically shows the same data as FIG. 44 plotted as the latetime apparent conductivity at t=1 second versus ratio of the late timeapparent conductivity at t=1 second over local environment conductivity;

FIG. 46 graphically shows distance to anomaly ahead of the tool versestransition time (t_(c)) as determined from the data of FIG. 36;

FIG. 47 shows a plot of the apparent conductivity (σapp(z; t)) in bothz- and t-coordinates;

FIG. 48 is a schematic showing apparent conductivity with a coplanartool;

FIG. 49 is a graph showing voltage response data in terms of theapparent conductivity (σ_(app)(t)) as a function of t as provided by thecoplanar tool of FIG. 48 at different distances from the bed;

FIG. 50 shows a comparison of the late time apparent conductivity(σ_(app)(t→∞)) for coplanar responses where d=05 m and L=01 m as afunction of conductivity of the local layer while the targetconductivity is fixed at 1 S/m;

FIG. 51 graphically shows the same data as FIG. 50 plotted as the ratioof target resistivity over local layer resistivity versus ratio of thelate time apparent conductivity over local layer resistivity; and

FIG. 52 graphically shows distance to anomaly ahead of the tool versestransition time (t_(c)) as determined from the data of FIG. 49;

FIG. 53 is a depiction of a two-layer salt dome profiling modelutilizing a coaxial tool;

FIG. 54 is a graph similar to FIG. 22, showing voltage response as afunction of t as given by the coaxial tool of FIG. 53 in a two-layerformation at different distances from a salt bed;

FIG. 55 schematically shows a model of a coaxial tool in a conductivelocal layer, a very resistive layer, and a further conductive layer;

FIG. 56 is a graph showing resistivity response versus time for ageometry as given in FIG. 55 for various thicknesses of the veryresistive layer;

FIG. 57 schematically shows a model of a coaxial tool in a resistivelocal layer, a conductive layer, and a further resistive layer;

FIG. 58 is a graph similar to FIG. 56, showing resistivity responseversus time for a geometry as given in FIG. 57 for various thicknessesof the conductive layer;

FIG. 59 schematically shows a model of a coaxial tool in a conductivelocal layer (1 ohm-m) in the vicinity of a highly resistive layer (100ohm-m) with a separating layer having an intermediate resistance (10ohm-m) of varying thickness in between;

FIG. 60 is a graph similar to FIG. 56, showing resistivity responseversus time for a geometry as given in FIG. 59 for various thicknessesof the separating layer;

FIG. 61 schematically shows a model of a structure involving a highlyresistive layer (100 ohm-m) covered by a conductive local layer (1ohm-m) which is covered by a resistive layer (10 ohm-m), whereby acoaxial tool is depicted in the resistive layer and the conductivelayer;

FIG. 62 on the left side shows apparent resistivity in both z and tcoordinates whereby inflection points are joined using curve fittedlines;

FIG. 62 on the right side shows an image log derived from the left side;

FIG. 63 on the right hand side schematically shows a coaxial tool seenas approaching a highly resistive formation at a dip angle ofapproximately 30 degrees;

FIG. 64 on the left hand side shows apparent dip response in both t andz coordinates for z-locations corresponding to those depicted in theright hand side.

DETAILED DESCRIPTION OF THE INVENTION

Embodiments of the invention relate to a system and method fordetermining distance and direction to an anomaly in a formation within awellbore. Both frequency domain excitation and time domain excitationhave been used to excite electromagnetic fields for use in anomalydetection. In frequency domain excitation, a device transmits acontinuous wave of a fixed or mixed frequency and measures responses atthe same band of frequencies. In time domain excitation, a devicetransmits a square wave signal, triangular wave signal, pulsed signal orpseudo-random binary sequence as a source and measures the broadbandearth response. Sudden changes in transmitter current cause signals toappear at a receiver caused by induction currents in the formation. Thesignals that appear at the receiver are called transient responsesbecause the receiver signals start at a first value and then decay orincrease with time to a constant level. The technique disclosed hereinimplements the time domain excitation technique.

As set forth below, embodiments of the invention propose a generalmethod to determine a direction to a resistive or conductive anomalyusing transient EM responses. As will be explained in detail, thedirection to the anomaly is specified by a dip angle and an azimuthangle. Embodiments of the invention propose to define an apparent dip(θ_(app)(t)) and an apparent azimuth (φ_(app)(t)) by combinations oftri-axial transient measurements. An apparent direction ({θ_(app)(t),φ_(app)(t)}) approaches a true direction ({θ, φ) as a time (t)increases. The θ_(app)(t) and φ_(app)(t) both initially read zero whenan apparent conductivity σ_(coaxial)(t) and σ_(coplanar)(t) from coaxialand coplanar measurements both read the conductivity around the tool.The apparent conductivity will be further explained below and can alsobe used to determine the location of an anomaly in a wellbore.

FIG. 1 illustrates a system that may be used to implement theembodiments of the method of the invention. A surface computing unit 10may be connected with an electromagnetic measurement tool 2 disposed ina wellbore 4 and supported by a cable 12. The cable 12 may beconstructed of any known type of cable for transmitting electricalsignals between the tool 2 and the surface computing unit 10. One ormore transmitters 16 and one are more receivers 18 may be provided fortransmitting and receiving signals. A data acquisition unit 14 may beprovided to transmit data to and from the transmitters 16 and receivers18 to the surface computing unit 10.

Each transmitter 16 and each receiver 18 may be tri-axial and therebycontain components for sending and receiving signals along each of threeaxes. Accordingly, each transmitter module may contain at least onesingle or multi-axis antenna and may be a 3-orthogonal componenttransmitter. Each receiver may include at least one single or multi-axiselectromagnetic receiving component and may be a 3-orthogonal componentreceiver.

The data acquisition unit 14 may include a controller for controllingthe operation of the tool 2. The data acquisition unit 14 preferablycollects data from each transmitter 16 and receiver 18 and provides thedata to the surface computing unit 10.

The surface computing unit 10 may include computer components includinga processing unit 30, an operator interface 32, and a tool interface 34.The surface computing unit 10 may also include a memory 40 includingrelevant coordinate system transformation data and assumptions 42, adirection calculation module 44, an apparent direction calculationmodule 46, and a distance calculation module 48. The surface computingunit 10 may further include a bus 50 that couples various systemcomponents including the system memory 40 to the processing unit 30. Thecomputing system environment 10 is only one example of a suitablecomputing environment and is not intended to suggest any limitation asto the scope of use or functionality of the invention. Furthermore,although the computing system 10 is described as a computing unitlocated on a surface, it may optionally be located below the surface,incorporated in the tool, positioned at a remote location, or positionedat any other convenient location.

The memory 40 preferably stores the modules 44, 46, and 48, which may bedescribed as program modules containing computer-executableinstructions, executed by the surface computing unit 10. The programmodule 44 contains the computer executable instruction necessary tocalculate a direction to an anomaly within a wellbore. The programmodule 46 includes the computer executable instructions necessary tocalculate an apparent direction as will be further explained below. Theprogram module 48 contains the computer executable instructionsnecessary to calculate a distance to an anomaly. The stored data 46includes data pertaining to the tool coordinate system and the anomalycoordinate system and other data required for use by the program modules44, 46, and 48. These program modules 44, 46, and 48, as well as thestored data 42, will be further described below in conjunction withembodiments of the method of the invention.

Generally, program modules include routines, programs, objects,components, data structures, etc. that perform particular tasks orimplement particular abstract data types. Moreover, those skilled in theart will appreciate that the invention may be practiced with othercomputer system configurations, including hand-held devices,multiprocessor systems, microprocessor-based or programmable consumerelectronics, minicomputers, mainframe computers, and the like. Theinvention may also be practiced in distributed computing environmentswhere tasks are performed by remote processing devices that are linkedthrough a communications network. In a distributed computingenvironment, program modules may be located in both local and remotecomputer storage media including memory storage devices.

Although the computing system 10 is shown as having a generalized memory40, the computing system 10 would typically includes a variety ofcomputer readable media. By way of example, and not limitation, computerreadable media may comprise computer storage media and communicationmedia. The computing system memory 40 may include computer storage mediain the form of volatile and/or nonvolatile memory such as a read onlymemory (ROM) and random access memory (RAM). A basic input/output system(BIOS), containing the basic routines that help to transfer informationbetween elements within computer 10, such as during start-up, istypically stored in ROM. The RAM typically contains data and/or programmodules that are immediately accessible to and/or presently beingoperated on by processing unit 30. By way of example, and notlimitation, the computing system 10 includes an operating system,application programs, other program modules, and program data.

The components shown in the memory 40 may also be included in otherremovable/nonremovable, volatile/nonvolatile computer storage media. Forexample only, a hard disk drive may read from or write to nonremovable,nonvolatile magnetic media, a magnetic disk drive may read from or writeto a removable, non-volatile magnetic disk, and an optical disk drivemay read from or write to a removable, nonvolatile optical disk such asa CD ROM or other optical media. Other removable/non-removable,volatile/non-volatile computer storage media that can be used in theexemplary operating environment include, but are not limited to,magnetic tape cassettes, flash memory cards, digital versatile disks,digital video tape, solid state RAM, solid state ROM, and the like. Thedrives and their associated computer storage media discussed above andillustrated in FIG. 1, provide storage of computer readableinstructions, data structures, program modules and other data for thecomputing system 10.

A user may enter commands and information into the computing system 10through input devices such as a keyboard and pointing device, commonlyreferred to as a mouse, trackball or touch pad. Input devices mayinclude a microphone, joystick, satellite dish, scanner, or the like.These and other input devices are often connected to the processing unit30 through the operator interface 32 that is coupled to the system bus50, but may be connected by other interface and bus structures, such asa parallel port or a universal serial bus (USB). A monitor or other typeof display device may be connected to the system bus 50 via aninterface, such as a video interface. In addition to the monitor,computers may also include other peripheral output devices such asspeakers and printer, which may be connected through an outputperipheral interface.

Although many other internal components of the computing system 10 arenot shown, those of ordinary skill in the art will appreciate that suchcomponents and the interconnection are well known. Accordingly,additional details concerning the internal construction of the computer10 need not be disclosed in connection with the present invention.

FIG. 2 is a flow chart illustrating the procedures involved in a methodof the invention. Generally, in procedure A, the transmitters 16transmit electromagnetic signals. In procedure B, the receivers 18receive transient responses. In procedure C, the system processes thetransient responses to determine a distance and direction to theanomaly.

FIGS. 3-6 illustrate the technique for implementing procedure C fordetermining distance and direction to the anomaly.

Tri-Axial Transient EM Responses

FIG. 3 illustrates directional angles between tool coordinates andanomaly coordinates. A transmitter coil T is located at an origin thatserves as the origin for each coordinate system. A receiver R is placedat a distance L from the transmitter. An earth coordinate system,includes a Z-axis in a vertical direction and an X-axis and a Y-axis inthe East and the North directions, respectively. The deviated boreholeis specified in the earth coordinates by a deviation angle θ_(b) and itsazimuth angle φ_(b). A resistivity anomaly A is located at a distance Dfrom the transmitter in the direction specified by a dip angle (θ_(a))and its azimuth (φ_(a)).

In order to practice embodiments of the method, FIG. 4A shows thedefinition of a tool/borehole coordinate system having x, y, and z axes.The z-axis defines the direction from the transmitter T to the receiverR. The tool coordinates in FIG. 4A are specified by rotating the earthcoordinates (X, Y, Z) in FIG. 3 by the azimuth angle (φ_(b)) around theZ-axis and then rotating by θ_(b) around the y-axis to arrive at thetool coordinates (x, y, z). The direction of the anomaly is specified bythe dip angle (Θ) and the azimuth angle (φ) where: $\begin{matrix}{{\cos\quad\vartheta} = {\left( {{\hat{b}}_{z} \cdot \hat{a}} \right) = {{\cos\quad\theta_{a}\cos\quad\theta_{b}} + {\sin\quad\theta_{a}\sin\quad\theta_{b}{\cos\left( {\varphi_{a} - \varphi_{b}} \right)}}}}} & (1) \\{{\tan\quad\phi} = \frac{\sin\quad\theta_{b}{\sin\left( {\varphi_{a} - \varphi_{b}} \right)}}{{\cos\quad\theta_{a}\sin\quad\theta_{b}{\cos\left( {\varphi_{a} - \varphi_{b}} \right)}} - {\sin\quad\theta_{a}\cos\quad\theta_{b}}}} & (2)\end{matrix}$

Similarly, FIG. 4B shows the definition of an anomaly coordinate systemhaving a, b, and c axes. The c-axis defines the direction from thetransmitter T to the center of the anomaly A. The anomaly coordinates inFIG. 4B are specified by rotating the earth coordinates (X, Y, Z) inFIG. 3 by the azimuth angle (φ_(a)) around the Z-axis and subsequentlyrotating by θ_(a) around the b-axis to arrive at the anomaly coordinates(a, b, c). In this coordinate system, the direction of the borehole isspecified in a reverse order by the azimuth angle (φ) and the dip angle(Θ).

Transient Responses in Two Coordinate Systems

The method is additionally based on the relationship between thetransient responses in two coordinate systems. The magnetic fieldtransient responses at the receivers [R_(x), R_(y), R_(z)] which areoriented in the [x, y, z] axis direction of the tool coordinates,respectively, are noted as $\begin{matrix}{\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix} = {\begin{bmatrix}R_{x} \\R_{y} \\R_{z}\end{bmatrix}\left\lbrack {M_{x}\quad M_{y}\quad M_{z}} \right\rbrack}} & (3)\end{matrix}$from a magnetic dipole source in each axis direction, [M_(x), M_(y),M_(z).]

When the resistivity anomaly is distant from the tool, the formationnear the tool is seen as a homogeneous formation. For simplicity, themethod may assume that the formation is isotropic. Only three non-zerotransient responses exist in a homogeneous isotropic formation. Theseinclude the coaxial response and two coplanar responses. Coaxialresponse V_(zz)(t) is the response when both the transmitter and thereceiver are oriented in the common tool axis direction. Coplanarresponses, V_(xx)(t) and V_(yy)(t), are the responses when both thetransmitter T and the receiver R are aligned parallel to each other buttheir orientation is perpendicular to the tool axis. All of thecross-component responses are identically zero in a homogeneousisotropic formation. Cross-component responses are either from alongitudinally oriented receiver with a transverse transmitter, or viseversa. Another cross-component response is also zero between a mutuallyorthogonal transverse receiver and transverse transmitter.

The effect of the resistivity anomaly is seen in the transient responsesas time increases. In addition to the coaxial and the coplanarresponses, the cross-component responses V_(ij)(t) (i≠j; i, j=x, y, z)become non-zero.

The magnetic field transient responses may also be examined in theanomaly coordinate system. The magnetic field transient responses at thereceivers [R_(a), R_(b), R_(c)] that are oriented in the [a, b, c] axisdirection of the anomaly coordinates, respectively, may be noted as$\begin{matrix}{\begin{bmatrix}V_{aa} & V_{ab} & V_{ac} \\V_{ba} & V_{bb} & V_{bc} \\V_{ca} & V_{cb} & V_{cc}\end{bmatrix} = {\begin{bmatrix}R_{a} \\R_{b} \\R_{c}\end{bmatrix}\left\lbrack {M_{a}\quad M_{b}\quad M_{c}} \right\rbrack}} & (4)\end{matrix}$from a magnetic dipole source in each axis direction, [M_(a), M_(b),M_(c)].

When the anomaly is large and distant compared to thetransmitter-receiver spacing, the effect of spacing can be ignored andthe transient responses can be approximated with those of the receiversnear the transmitter. Then, the method assumes that axial symmetryexists with respect to the c-axis that is the direction from thetransmitter to the center of the anomaly. In such an axially symmetricconfiguration, the cross-component responses in the anomaly coordinatesare identically zero in time-domain measurements. $\begin{matrix}{\begin{bmatrix}V_{aa} & V_{ab} & V_{ac} \\V_{ba} & V_{bb} & V_{bc} \\V_{ca} & V_{cb} & V_{cc}\end{bmatrix} = \begin{bmatrix}V_{aa} & 0 & 0 \\0 & V_{aa} & 0 \\0 & 0 & V_{cc}\end{bmatrix}} & (5)\end{matrix}$

The magnetic field transient responses in the tool coordinates arerelated to those in the anomaly coordinates by a simple coordinatetransformation P(Θ, φ) specified by the dip angle (Θ) and azimuth angle(φ). $\begin{matrix}{\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix} = {{{P\left( {\vartheta,\phi} \right)}^{tr}\begin{bmatrix}V_{aa} & V_{ab} & V_{ac} \\V_{ba} & V_{bb} & V_{bc} \\V_{ca} & V_{cb} & V_{cc}\end{bmatrix}}{P\left( {\vartheta,\phi} \right)}}} & (6) \\{{P\left( {\vartheta,\phi} \right)} = \begin{bmatrix}{\cos\quad\vartheta\quad\cos\quad\phi} & {\cos\quad\vartheta\quad\sin\quad\phi} & {{- \sin}\quad\vartheta} \\{{- \sin}\quad\phi} & {\cos\quad\phi} & 0 \\{\sin\quad\vartheta\quad\cos\quad\phi} & {\sin\quad\vartheta\quad\sin\quad\phi} & {\cos\quad\vartheta}\end{bmatrix}} & (7)\end{matrix}$Determination of Target Direction

The assumptions set forth above contribute to determination of targetdirection, which is defined as the direction of the anomaly from theorigin. When axial symmetry in the anomaly coordinates is assumed, thetransient response measurements in the tool coordinates are constrainedand the two directional angles may be determined by combinations oftri-axial responses. $\begin{matrix}{\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix} = {{{P\left( {\vartheta,\phi} \right)}^{tr}\begin{bmatrix}V_{aa} & 0 & 0 \\0 & V_{aa} & 0 \\0 & 0 & V_{cc}\end{bmatrix}}{P\left( {\vartheta,\phi} \right)}}} & (8)\end{matrix}$

In terms of each tri-axial responseV _(xx)=(V _(aa) cos² Θ+V _(cc) sin²Θ)cos² φ+V _(aa) sin²φV _(yy)=(V _(aa) cos² Θ+V _(cc) sin²Θ)sin² φ+V _(aa) cos²φV _(zz) =V _(aa) sin² Θ+V _(cc) cos²Θ  (9)V _(xy) =V _(yx)=−(V _(aa) −V _(cc))sin²Θ cos φ sin φV _(zx) =V _(xz)=−(V _(aa) −V _(cc))cos Θ sin Θ cos φV _(yz) =V _(zy)=−(V _(aa) −V _(cc))cos Θ sin Θ sin φ  (10)

The following relations can be noted:V _(xx) +V _(yy) +V _(zz)=2V _(aa) +V _(cc)V _(xx) −V _(yy)=(V _(cc) −V _(aa))sin²Θ (cos²φ−sin²φ)V _(yy) −V _(zz)−(V _(cc) −V _(aa))(cos²Θ−sin²Θ sin²φ)V _(zz) −V _(xx)=(V _(cc) −V _(aa))(cos²Θ−sin²Θ cos²φ)   (11)

Several distinct cases can be noted. In the first of these cases, whennone of the cross-components is zero, V_(xy)≠0 nor V_(yz)≠0 norV_(zx)≠0, then the azimuth angle φ is not zero nor π/2 (90°), and can bedetermined by, $\begin{matrix}{{\phi = {\frac{1}{2}\tan^{- 1}\frac{V_{xy} + V_{yx}}{V_{xx} - V_{yy}}}}{\phi = {{\tan^{- 1}\frac{V_{yz}}{V_{xz}}} = {\tan^{- 1}\frac{V_{zy}}{V_{zx}}}}}} & (12)\end{matrix}$

By noting the relation, $\begin{matrix}{\frac{V_{xy}}{V_{xz}} = {{\tan\quad\vartheta\quad\sin\quad\phi\quad{and}\frac{V_{xy}}{V_{yz}}} = {\tan\quad\vartheta\quad\cos\quad\phi}}} & (13)\end{matrix}$the dip (deviation) angle Θ is determined by, $\begin{matrix}{{\tan\quad\vartheta} = \sqrt{\left( \frac{V_{xy}}{V_{xz}} \right)^{2} + \left( \frac{V_{xy}}{V_{yz}} \right)^{2}}} & (14)\end{matrix}$

In the second case, when V_(xy)=0 and V_(yz)=0, then Θ=0 or φ=0 or π(180°) or φ=±π/2 (90°) and Θ=±π/2 (90°), as the coaxial and the coplanarresponses should differ from each other (V_(aa)≠V_(cc)).

If ≠=0, then the dip angle Θ is determined by, $\begin{matrix}{\vartheta = {{- \frac{1}{2}}\tan^{- 1}\frac{V_{xz} + V_{zx}}{V_{xx} - V_{zz}}}} & (15)\end{matrix}$

If φ=π (180°), then the dip angle Θ is determined by, $\begin{matrix}{\vartheta = {{+ \frac{1}{2}}\tan^{- 1}\frac{V_{xz} + V_{zx}}{V_{xx} - V_{zz}}}} & (16)\end{matrix}$

Also, with regard to the second case, If Θ=0, then V_(xx)=V_(yy) andV_(zx)=0. If φ=±π/2 (90°) and Θ=±π/2 (90°), then V_(zz)=V_(xx) andV_(zx)=0. These instances are further discussed below with relation tothe fifth case.

In the third case, when V_(xy)=0 and V_(xz)=0, then φ=±π/2 (90°) or Θ=0or φ=0 and Θ=±π/2 (90°).

If φ=π/2, then the dip angle Θ is determined by, $\begin{matrix}{\vartheta = {{- \frac{1}{2}}\tan^{- 1}\frac{V_{yz} + V_{zy}}{V_{yy} - V_{zz}}}} & (17)\end{matrix}$

If φ=−π/2, then the dip angle Θ is determined by, $\begin{matrix}{\partial{= {{+ \frac{1}{2}}\tan^{- 1}\frac{V_{yz} + V_{zy}}{V_{yy} - V_{zz}}}}} & (18)\end{matrix}$

Also with regard to the third case, If Θ=0, then V_(xx)=V_(yy) andV_(yz)=0. If φ=0 and Θ=±π/2 (90°), V_(yy)=V_(zz) and V_(yz)=0. Thesesituations are further discussed below with relation to the fifth case.

In the fourth case, V_(xz)=0 and V_(yz)=0, then Θ=0 or π(18°) or ±π/2(90°).

If Θ=±π/2, then the azimuth angle φ is determined by, $\begin{matrix}{\phi = {{- \frac{1}{2}}\tan^{- 1}\frac{V_{xy} + V_{yx}}{V_{xx} - V_{yy}}}} & (19)\end{matrix}$

Also with regard to the fourth case, if Θ=0 or π(180°), thenV_(xx)=V_(yy) and V_(yz)=0. This situation is also shown below withrelation to the fifth case.

In the fifth case, all cross components vanish, V_(xz)=V_(yz)=V_(xy)=0then Θ=0, or Θ=±π/2 (90°) and φ=0 or ±π/2 (90°).

If V_(xx)=V_(yy) then Θ=0 or π(180°).

If V_(yy)=V_(zz) then Θ=±π/2 (90°) and φ=0.

If V_(zz)=V_(xx) then Θ=±π/2 (90°) and φ=±π/2 (90°).

Tool rotation around the tool/borehole axis

In the above analysis, all the transient responses V_(ij)(t) (i, j=x, y,z) are specified by the x-, y-, and z-axis directions of the toolcoordinates. However, the tool rotates inside the borehole and theazimuth orientation of the transmitter and the receiver no longercoincides with the x- or y-axis direction as shown in FIG. 5. If themeasured responses are {tilde over (V)}_({tilde over (ij)})(ĩ,{tildeover (j)}={tilde over (x)} and {tilde over (y)} axis are the directionof antennas fixed to the rotating tool, and ψ is the tool's rotationangle, then $\begin{matrix}{\begin{bmatrix}V_{\overset{\sim}{x}\overset{\sim}{x}} & V_{\overset{\sim}{x}\overset{\sim}{y}} & V_{\overset{\sim}{x}z} \\V_{\overset{\sim}{y}\overset{\sim}{x}} & V_{\overset{\sim}{y}\overset{\sim}{y}} & V_{\overset{\sim}{y}z} \\V_{z\overset{\sim}{x}} & V_{z\overset{\sim}{y}} & V_{zz}\end{bmatrix} = {{{R(\psi)}^{tr}\begin{bmatrix}V_{xx} & V_{xy} & V_{xz} \\V_{yx} & V_{yy} & V_{yz} \\V_{zx} & V_{zy} & V_{zz}\end{bmatrix}}{R(\psi)}}} & (20) \\{{{R(\psi)} = \begin{bmatrix}{\cos\quad\psi} & {{- \sin}\quad\psi} & 0 \\{\sin\quad\psi} & {\cos\quad\psi} & 0 \\0 & 0 & 1\end{bmatrix}}{{Then},}} & (21) \\{{{V_{\overset{\sim}{x}\overset{\sim}{x}} = {{\left( {{V_{aa}\cos^{2}\vartheta} + {V_{cc}\sin^{2}\vartheta}} \right){\cos^{2}\left( {\phi - \psi} \right)}} + {V_{aa}{\sin^{2}\left( {\phi - \psi} \right)}}}}V_{\overset{\sim}{y}\overset{\sim}{y}} = {{\left( {{V_{aa}\cos^{2}\vartheta} + {V_{cc}\sin^{2}\vartheta}} \right){\sin^{2}\left( {\phi - \psi} \right)}} + {V_{aa}{\cos^{2}\left( {\phi - \psi} \right)}}}}{V_{zz} = {{V_{aa}\sin^{2}\vartheta} + {V_{cc}\cos^{2}\vartheta}}}} & (22) \\{{V_{\overset{\sim}{x}\overset{\sim}{y}} = {V_{\overset{\sim}{y}\overset{\sim}{x}} = {{- \left( {V_{aa} - V_{cc}} \right)}\sin^{2}\vartheta\quad{\cos\left( {\phi - \psi} \right)}{\sin\left( {\phi - \psi} \right)}}}}{V_{z\overset{\sim}{x}} = {V_{\overset{\sim}{x}z} = {{- \left( {V_{aa} - V_{cc}} \right)}\cos\quad\vartheta\quad\sin\quad\vartheta\quad{\cos\left( {\phi - \psi} \right)}}}}{V_{\overset{\sim}{y}z} = {V_{z\overset{\sim}{y}} = {{- \left( {V_{aa} - V_{cc}} \right)}\cos\quad\vartheta\quad\sin\quad\vartheta\quad{\sin\left( {\phi - \psi} \right)}}}}} & (23)\end{matrix}$

The following relations apply:V _({tilde over (xx)}) +V _({tilde over (yy)}) +V _(zz)=2V _(aa) +V_(cc)V _({tilde over (xx)}) −V _({tilde over (yy)})=(V _(cc) −V _(aa))sin²Θ{²(φ−ψ)−sin²(φ−ψ)}V _({tilde over (yy)}) −V _(zz)=−(V _(cc) −V _(aa)){ cos²Θ−sin²Θsin²(φ−ψ)}V _(zz) −V _({tilde over (xx)})=(V _(cc) −V _(aa)){ cos²Θ−sin²Θcos²(φ−ψ)}  (24)

Consequently, $\begin{matrix}{{{\phi - \psi} = {\frac{1}{2}\tan^{- 1}\frac{V_{\overset{\sim}{x}\overset{\sim}{y}} + V_{\overset{\sim}{y}\overset{\sim}{x}}}{V_{\overset{\sim}{x}\overset{\sim}{x}} - V_{\overset{\sim}{y}\overset{\sim}{y}}}}}{{\phi - \psi} = {{\tan^{- 1}\frac{V_{\overset{\sim}{y}z}}{V_{\overset{\sim}{x}z}}} = {\tan^{- 1}\frac{V_{z\overset{\sim}{y}}}{V_{z\overset{\sim}{x}}}}}}} & (25)\end{matrix}$

The azimuth angle φ is measured from the tri-axial responses if the toolrotation angle ψ is known. To the contrary, the dip (deviation) angle Θis determined by $\begin{matrix}{{\tan\quad\vartheta} = \sqrt{\left( \frac{V_{\overset{\sim}{x}\overset{\sim}{y}}}{V_{\overset{\sim}{x}z}} \right)^{2} + \left( \frac{V_{\overset{\sim}{x}\overset{\sim}{y}}}{V_{\overset{\sim}{y}z}} \right)^{2}}} & (26)\end{matrix}$without knowing the tool orientation ψ. Apparent dip angle and azimuthangle and the distance to the anomaly The dip and the azimuth angledescribed above indicate the direction of a resistivity anomalydetermined by a combination of tri-axial transient responses at a time(t) when the angles have deviated from a zero value. When t is small orclose to zero, the effect of such anomaly is not apparent in thetransient responses as all the cross-component responses are vanishing.To identify the anomaly and estimate not only its direction but also thedistance, it is useful to define the apparent azimuth angle φ_(app)(t)by, $\begin{matrix}{{{\phi_{app}(t)} = {\frac{1}{2}\tan^{- 1}\frac{{V_{xy}(t)} + {V_{yx}(t)}}{{V_{xx}(t)} - {V_{yy}(t)}}}}{{\phi_{app}(t)} = {{\tan^{- 1}\frac{V_{yz}(t)}{V_{xz}(t)}} = {\tan^{- 1}\frac{V_{zy}(t)}{V_{zx}(t)}}}}} & (27)\end{matrix}$and the effective dip angle Θ_(app)(t) by $\begin{matrix}{{\tan\quad{\vartheta_{app}(t)}} = \sqrt{\left( \frac{V_{xy}(t)}{V_{xz}(t)} \right)^{2} + \left( \frac{V_{xy}(t)}{V_{yz}(t)} \right)^{2}}} & (28)\end{matrix}$for the time interval when φ_(app)(t)≠0 nor π/2 (90°). For simplicity,the case examined below is one in which none of the cross-componentmeasurements is identically zero: V_(xy)(t)≠0 V_(yz)(t)≠0, andV_(zx)(t)≠0.For the time interval when φ_(app)(t)=0, Θ_(app)(t) is defined by,$\begin{matrix}{{\vartheta_{app}(t)} = {{- \frac{1}{2}}\tan^{- 1}\frac{{V_{xz}(t)} + {V_{zx}(t)}}{{V_{xx}(t)} - {V_{zz}(t)}}}} & (29)\end{matrix}$For the time interval when φ_(app)(t)=π/2 (90°), Θ_(app)(t) is definedby, $\begin{matrix}{{\vartheta_{app}(t)} = {{- \frac{1}{2}}\tan^{- 1}\frac{{V_{yz}(t)} + {V_{zy}(t)}}{{V_{yy}(t)} - {V_{zz}(t)}}}} & (30)\end{matrix}$

When t is small and the transient responses do not see the effect of aresistivity anomaly at distance, the effective angles are identicallyzero, Θ_(app)(t)=Θ_(app)(t)=0. As t increases, when the transientresponses see the effect of the anomaly, φ_(app)(t) and Θ_(app)(t) beginto show the true azimuth and the true dip angles. The distance to theanomaly may be indicated at the time when φ_(app)(t) and Θ_(app)(t)start deviating from the initial zero values. As shown below in amodeling example, the presence of an anomaly is detected much earlier intime in the effective angles than in the apparent conductivity(σ_(app)(t)). Even if the resistivity of the anomaly may not be knownuntil σ_(app)(t) is affected by the anomaly, its presence and thedirection can be measured by the apparent angles. With limitation intime measurement, the distant anomaly may not be seen in the change ofσ_(app)(t) but is visible in φ_(app)(t) and Θ_(app)(t).

MODELING EXAMPLE

FIG. 6 depicts a simplified modeling example wherein a resistivityanomaly A is a massive salt dome, and the salt interface 55 may beregarded as a plane interface. For further simplification, it can beassumed that the azimuth of the salt face is known. Accordingly, theremaining unknowns are the distance D to the salt face from the tool,the isotropic or anisotropic formation resistivity, and the approachangle (or dip angle) θ as shown in FIG. 6. FIG. 6 also indicates coaxial(60) coplanar (62), and cross-component (64) measurement arrangements.

FIG. 7 and FIG. 8 below show the voltage from the coaxial V_(zz)(t),coplanar V_(xx)(t), and the cross-component V_(zx)(t) measurements forL=1 m, for θ=30°, and at salt distance D=10 m and D=100 m respectively.The apparent dip θ_(app)(t) is defined by, $\begin{matrix}{{\theta_{app}(t)} = {{- \frac{1}{2}}\tan^{- 1}{\frac{{V_{zx}(t)} + {V_{xz}(t)}}{{V_{zz}(t)} - {V_{xx}(t)}}.}}} & (31)\end{matrix}$

FIG. 9 shows the apparent dip (θ_(app)(t)) for the L=1 m tool assemblywhen the salt face is D=10 m away and at the approach angle of θ=30°.

In addition, the apparent conductivity (σ_(app)(t)) from both thecoaxial (V_(zz)(t)) and the coplanar (V_(xx)(t)) responses is shown inFIG. 10, wherein the approach angle (θ) and salt face distance (D) arethe same as in FIG. 9.

Also plotted is the ratio, σ_(app-coplanar)(t)/σ_(app-coaxial)(t), thatis available without cross-component V_(zx)(t) measurements as shown inFIG. 11, wherein the approach angle (θ) and salt face distance (D) arethe same as in FIG. 3.

Note that the direction to the salt face is immediately identified inthe apparent dip θ_(app)(t) plot of FIG. 9 as early as 10⁻⁴ second whenthe presence of the resistivity anomaly is barely detected in theapparent conductivity (σ_(app)(t)) plot of FIG. 10. It takes almost 10⁻³second for the apparent conductivity to approach an asymptoticσapp(later t) value and for the apparent conductivity ratio to readθ=30°.

FIG. 12 shows the apparent dip θapp(t) for the L=1 m tool assembly whenthe salt face is D=10 m away, but at different angles between the toolaxis and the target. The approach angle (θ) may be identified at anyangle.

FIG. 12 and FIGS. 13 and 14 compare the apparent dip θ_(app)(t) fordifferent salt face distances (D) and different angles between the toolaxis and the target.

The distance to the salt face can be also determined by the transitiontime at which θ_(app)(t) takes an asymptotic value. Even if the saltface distance (D) is 100 m, it can be identified and its direction canbe measured by the apparent dip θ_(app)(t).

In summary, the method considers the coordinate transformation oftransient EM responses between tool-fixed coordinates and anomaly-fixedcoordinates. When the anomaly is large and far away compared to thetransmitter-receiver spacing, one may ignore the effect of spacing andapproximate the transient EM responses with those of the receivers nearthe transmitter. Then, one may assume axial symmetry exists with respectto the c-axis that defines the direction from the transmitter to theanomaly. In such an axially symmetric configuration, the cross-componentresponses in the anomaly-fixed coordinates are identically zero. Withthis assumption, a general method is provided for determining thedirection to the resistivity anomaly using tri-axial transient EMresponses.

The method defines the apparent dip θ_(app)(t) and the apparent azimuthφ_(app)(t) by combinations of tri-axial transient measurements. Theapparent direction {θ_(app)(t), φ_(app)(t)} reads the true direction {θ,φ} at later time. The θ_(app)(t) and φ_(app)(t) both read zero when t issmall and the effect of the anomaly is not sensed in the transientresponses or the apparent conductivity. The conductivities(σ_(coaxial)(t) and σ_(coplanar)(t)) from the coaxial and coplanarmeasurements both indicate the conductivity of the near formation aroundthe tool.

Deviation of the apparent direction ({θ_(app)(t), φ_(app)(t)}) from zeroidentifies the anomaly. The distance to the anomaly is measured by thetime when the apparent direction ({σ_(app)(t), φ_(app)(t)}) approachesthe true direction ({θ, φ}). The distance can be also measured from thechange in the apparent conductivity. However, the anomaly is identifiedand measured much earlier in time in the apparent direction than in theapparent conductivity.

Apparent Conductivity

As set forth above, apparent conductivity can be used as an alternativetechnique to apparent angles in order to determine the location of ananomaly in a wellbore. The time-dependent apparent conductivity can bedefined at each point of a time series at each logging depth. Theapparent conductivity at a logging depth z is defined as theconductivity of a homogeneous formation that would generate the sametool response measured at the selected position.

In transient EM logging, transient data are collected at a logging depthor tool location z as a time series of induced voltages in a receiverloop. Accordingly, time dependent apparent conductivity (σ (z; t)) maybe defined at each point of the time series at each logging depth, for aproper range of time intervals depending on the formation conductivityand the tool specifications.

Apparent Conductivity for a Coaxial Tool

The induced voltage of a coaxial tool with transmitter-receiver spacingL in the homogeneous formation of conductivity (σ) is given by,$\begin{matrix}{{V_{zZ}(t)} = {{C\frac{\left( {\mu_{0}\sigma} \right)^{3/2}}{8t^{5/2}}e^{- u^{2}}\quad{where}\quad u^{2}} = {\frac{\mu_{0}\sigma}{4}\frac{L^{2}}{t}}}} & (32)\end{matrix}$and C is a constant.

FIG. 15 illustrates a coaxial tool in which both a transmitter coil (T)and a receiver coil (R) are wound around the common tool axis. Thesymbols σ₁ and σ₂ may represent the conductivities of two formationlayers. This tool is used to illustrate the voltage response fordifferent values of t and L in FIGS. 17-18 below, where σ₁=σ₂.

FIG. 16 shows the voltage response of the coaxial tool with L=01 m in ahomogeneous formation for various formation resistivities (R) from 1000ohm-m to 0.1 ohm-m. The voltage is positive at all times t for t>0. Theslope of the voltage is nearly constant$\frac{{\partial\ln}\quad{V_{zZ}(t)}}{{\partial\ln}\quad t} \approx {- \frac{5}{2}}$in the time interval between 10⁻⁸ second and 1 second (and later) forany formation resistivity larger than 10 ohm-m. The slope changes signat an earlier time around 10⁻⁶ second when the resistivity is low as 0.1ohm-m.

FIG. 17 shows the voltage response as a function of formationresistivity at different times (t) for the same coaxial tool spacing(L=1 m). For the resistivity range from 0.1 ohm-m to 100 ohm-m, thevoltage response is single valued as a function of formation resistivityfor the measurement time (t) later than 10⁻⁶ second. At smaller times(t), for instance at 10⁻⁷ second, the voltage is no longer singlevalued. The same voltage response is realized at two different formationresistivity values.

FIG. 18 shows the voltage response as a function of formationresistivity for a larger transmitter-receiver spacing of L=10 m on acoaxial tool. The time interval when the voltage response is singlevalued is shifted toward larger times (t). The voltage response issingle valued for resistivity from 0.1 ohm-m to 100 ohm-m, for themeasurement time (t) later than 10⁻⁴ second. At smaller values of t, forinstance at t=10⁻⁵ second, the voltage is no longer single valued. Theapparent conductivity from a single measurement (coaxial, singlespacing) alone is not well defined.

For relatively compact transmitter-receiver spacing (L=1 m to 10 m), andfor the time measurement interval where t is greater than 10⁻⁶second,the transient EM voltage response is mostly single valued as a functionof formation resistivity between 0.1-ohm-m and 100 ohm-m (and higher).This enables definition of the time-changing apparent conductivity fromthe voltage response (V_(zZ)(t)) at each time of measurement as:$\begin{matrix}{{C\frac{\left( {\mu_{0}{\sigma_{app}(t)}} \right)^{3/2}}{8\quad t^{5/2}}e^{- {u_{app}{(t)}}^{2}}} = {{{V_{zZ}(t)}\quad{where}\quad{u_{app}(t)}^{2}} = {\frac{\mu_{0}{\sigma_{app}(t)}}{4}\frac{L^{2}}{t}}}} & (33)\end{matrix}$and V_(zZ)(t) on the right hand side is the measured voltage response ofthe coaxial tool. From a single type of measurement (coaxial, singlespacing), the greater the spacing L, the larger the measurement time (t)should be to apply the apparent conductivity concept. The σ_(app)(t)should be constant and equal to the formation conductivity in ahomogeneous formation: σ_(app)(t)=σ. The deviation from a constant (σ)at time (t) suggests a conductivity anomaly in the region specified bytime (t).Apparent Conductivity for a Coplanar Tool

The induced voltage of the coplanar tool with transmitter-receiverspacing L in the homogeneous formation of conductivity (σ) is given by,$\begin{matrix}{{V_{xX}(t)} = {{C\frac{\left( {\mu_{0}\sigma} \right)^{3/2}}{8\quad t^{5/2}}\left( {1 - u^{2}} \right)e^{- u^{2}}\quad{where}\quad u^{2}} = {\frac{\mu_{0}\sigma}{4t}L^{2}}}} & (34)\end{matrix}$and C is a constant. At small values of t, the 4t coplanar voltagechanges polarity depending on the spacing L and the formationconductivity.

FIG. 19 illustrates a coplanar tool in which the transmitter (T) and thereceiver (R) are parallel to each other and oriented perpendicularly tothe tool axis. The symbols σ₁ and σ₂ may represent the conductivities oftwo formation layers. This tool is used to illustrate the voltageresponse for different values of t and L in FIGS. 21-22 below, whereσ₁=σ₂.

FIG. 20 shows the voltage response of a coplanar tool with a length L=1m as a function of formation resistivity at different times (t). For theresistivity range from 0.1 ohm-m to 100 ohm-m, the voltage response issingle valued as a function of formation resistivity for values of tlarger than 10⁻⁶ second. At smaller values of t, for instance at t=10⁻⁷second, the voltage changes polarity and is no longer single valued.

FIG. 21 shows the voltage response as a function of formationresistivity at different times (t) for a longer coplanar tool with alength L=5 m. The time interval when the voltage response is singlevalued is shifted towards larger values of t.

Similarly to the coaxial tool response, the time-changing apparentconductivity is defined from the coplanar tool response V_(xX)(t) ateach time of measurement as, $\begin{matrix}{{C\frac{\left( {\mu_{0}{\sigma_{app}(t)}} \right)^{3/2}}{8t^{5/2}}\left( {1 - {u_{app}(t)}^{2}} \right)e^{- {u_{app}{(t)}}^{2}}} = {{{V_{xX}(t)}\quad{where}\quad{u_{app}(t)}^{2}} = {\frac{\mu_{0}{\sigma_{app}(t)}}{4}\frac{L^{2}}{t}}}} & (35)\end{matrix}$and V_(xX)(t) on the right hand side is the measured voltage response ofthe coplanar tool. The longer the spacing, the larger the value t shouldbe to apply the apparent conductivity concept from a single type ofmeasurement (coplanar, single spacing). The σ_(app)(t) should beconstant and equal to the formation conductivity in a homogeneousformation: σ_(app)(t)=σ.Apparent Conductivity For a Pair of Coaxial Tools

When there are two coaxial receivers, the ratio between the pair ofvoltage measurements is given by, $\begin{matrix}{\frac{V_{zZ}\left( {L_{1};t} \right)}{V_{zZ}\left( {L_{2};t} \right)} = e^{{- \frac{\mu_{0}\sigma}{4\quad t}}{({L_{2}^{1} - L_{2}^{2}})}}} & (36)\end{matrix}$where L₁ and L₂ are transmitter-receiver spacing of two coaxial tools.

Conversely, the time-changing apparent conductivity is defined for apair of coaxial tools by, $\begin{matrix}{{\sigma_{app}(t)} = {\frac{- {\ln\left( \frac{V_{zZ}\left( {L_{1};t} \right)}{V_{zZ}\left( {L_{2};t} \right)} \right)}}{\left( {L_{1}^{2} - L_{2}^{2}} \right)}\frac{4t}{\mu_{0}}}} & (37)\end{matrix}$at each time of measurement. The σ_(app)(t) should be constant and equalto the formation conductivity in a homogeneous formation: σ_(app)(t)=σ.

The apparent conductivity is similarly defined for a pair of coplanartools or for a pair of coaxial and coplanar tools. The σ_(app)(t) shouldbe constant and equal to the formation conductivity in a homogeneousformation: σ_(app)(t)=σ. The deviation from a constant (σ) at time (t)suggests a conductivity anomaly in the region specified by time (t).

Analysis of Coaxial Transient Response in Two-layer Models

To illustrate usefulness of the concept of apparent conductivity, thetransient response of a tool in a two-layer earth model, as in FIG. 15for example, can be examined. A coaxial tool with a transmitter-receiverspacing L may be placed in a horizontal well. Apparent conductivity(σ_(app)(t)) reveals three parameters including:

(1) the conductivity ( in the present example σ₁=0.1 S/m) of a firstlayer in which the tool is placed; the conductivity (in the presentexample σ₂=1 S/m) of an adjacent bed; and

(2) the distance of the tool (horizontal borehole) to the layerboundary, for which in the present example d=1, 5, 10, 25, and 50 m areshown.

Under a more general circumstance, the relative direction of a boreholeand tool to the bed interface is not known. In the case of horizontalwell logging, it's trivial to infer that the tool is parallel to theinterface as the response does not change when the tool moves.

The voltage response of the L=01 m transmitter-receiver offset coaxialtool at different distances is shown in FIG. 22. Information can bederived from these responses using apparent conductivity as furtherexplained with regard to FIG. 23. FIG. 23 shows the voltage data of FIG.22 plotted in terms of apparent conductivity. The apparent conductivityplot shows conductivity at small t, conductivity at large t, and thetransition time that moves as the distance (d) changes.

As will be further explained below, in a two-layer resistivity profile,the apparent conductivity as t approaches zero can identify the layerconductivity around the tool, while the apparent conductivity as tapproaches infinity can be used to determine the conductivity of theadjacent layer at a distance. The distance to a bed boundary from thetool can also be measured from the transition time observed in theapparent conductivity plot. The apparent conductivity plot for both timeand tool location may be used as an image presentation of the transientdata. Similarly, FIG. 24 illustrates the apparent conductivity in atwo-layer model where σ₁=1 S/m (R₁=1 ohm-m) and σ₂=0.1 S/m (R₂=1ohm-m)).

Conductivity at Small Values of t

At small values of t, the tool reads the apparent conductivity of thefirst layer around the tool. At large values of t, the tool reads 0.4S/m for a two-layer model where σ₁=0.1 S/m (R₁=10 ohm-m) and σ₂=1 S/m(R₂=1 ohm-m), which is an average between the conductivities of the twolayers. The change of distance (d) is reflected in the transition time.

Conductivity at small values of t is the conductivity of the local layerwhere the tool is located. At small values of t, the signal reaches thereceiver directly from the transmitter without interfering with the bedboundary. Namely, the signal is affected only by the conductivity aroundthe tool. Conversely, the layer conductivity can be measured easily byexamining the apparent conductivity at small values of t.

Conductivity at Large Values of t

Conductivity at large values of t is some average of conductivities ofboth layers. At large values of t, nearly half of the signals come fromthe formation below the tool and the remaining signals come from above,if the time for the signal to travel the distance between the tool andthe bed boundary is small.

FIG. 25 compares the σ_(app)(t) plot of FIGS. 23 and 25 for L=1 m andd=1 m where the resistivity ratio R₁/R₂ is 10:1 in FIG. 23 and 1:10 inFIG. 24. Though not shown, the conductivity at large values of t has aslight dependence on d. When the dependence is ignored, the conductivityat large values of t is determined solely by the conductivities of thetwo layers and is not affected by the location of the tool in layer 1 orlayer 2.

FIG. 26 compares the σapp(t) plots for d=1 m but with different spacingsL. The σapp(t) reaches the nearly constant conductivity at large valuesof t as L increases. However, the conductivity at large values of t isalmost independent of the spacing L for the range of d and theconductivities considered.

FIG. 27 compares the σapp(t) plots for d=1 m and L=1 m but for differentresistivity ratios. The apparent conductivity at large t is proportionalto σ₁ for the same ratio (σ₁/σ₂). For instance: $\begin{matrix}{{\sigma_{app}\left( {{\left. t\rightarrow\infty \right.;{{R_{1}/R_{2}} = 10}},{R_{1} = {10\quad{ohm}\text{-}m}}} \right)} = {10*{\sigma_{app}\left( {{\left. t\rightarrow\infty \right.;{{R_{1}/R_{2}} = 10}},{R_{1} = {100\quad{ohm}\text{-}m}}} \right)}}} & (38)\end{matrix}$

FIG. 28 shows examples of the app (t) plots for d=1 m and L=1 m but fordifferent resistivity ratios of the target layer 2 while the localconductivity (σ₁) is fixed at 1 S/m (R₁=1 ohm-m). The apparentconductivity at large values of t is determined by the target layer 2conductivity, as shown in FIG. 29 when σ₁ is fixed at 1 S/m.

Numerically, the late time conductivity may be approximated by thesquare root average of two-layer conductivities as: $\begin{matrix}{\sqrt{\sigma_{app}\left( {{{t->\infty};\sigma_{1}},\sigma_{2}} \right)} = \frac{\sqrt{\sigma_{1}} + \sqrt{\sigma_{2}}}{2}} & (39)\end{matrix}$

To summarize, the conductivity at large values of t (as t approachesinfinity) can be used to estimate the conductivity (σ₂) of the adjacentlayer when the local conductivity (σ₁) near the tool is known, forinstance from the conductivity as t approaches 0 as illustrated in FIG.30.

Estimation of d, The Distance to the Adjacent Bed

The transition time at which the apparent conductivity (σapp(t)) startsdeviating from the local conductivity (σ1) towards the conductivity atlarge values of t depends on d and L, as shown in FIG. 31. Forconvenience, the transition time (tc) can be defined as the time atwhich the σapp(tc) takes the cutoff conductivity (σc). In this case, thecutoff conductivity is represented by the arithmetic average between theconductivity as t approaches zero and the conductivity as t approachesinfinity. The transition time (t_(c)) is dictated by the ray path:$\begin{matrix}{\sqrt{\left( \frac{L}{2} \right)^{2} + d^{2}},} & (40)\end{matrix}$that is the shortest distance for the EM signal traveling from thetransmitter to the bed boundary, to the receiver, independently of theresistivity of the two layers. Conversely, the distance (d) can beestimated from the transition time (t_(c)), as shown in FIG. 32.Other Uses of Apparent Conductivity

Similarly to conventional induction tools, the apparent conductivity(σ_(app)(Z)) is useful for analysis of the error in transient signalprocessing. The effect of the noise in transient response data may beexamined as the error in the conductivity determination. A plot of theapparent conductivity (σ_(app)(z; t)) for different distances (d) inboth the z and t coordinates may serve as an image presentation of thetransient data as shown in FIG. 33 for a L=1 m tool. The z coordinatereferences the tool depth along the borehole. The σ_(app)(z; t) plotshows the approaching bed boundary as the tool moves along the borehole.

The apparent conductivity should be constant and equal to the formationconductivity in a homogeneous formation. The deviation from a constantconductivity value at time (t) suggests the presence of a conductivityanomaly in the region specified by time (t).

Look-Ahead Capabilities of EM Transient Method

By analyzing apparent conductivity or its inherent inverse equivalent(apparent resistivity), the present invention can identify the locationof a resistivity anomaly (e.g., a conductive anomaly and a resistiveanomaly). Further, resistivity or conductivity can be determined fromthe coaxial and/or coplanar transient responses. As explained above, thedirection of the anomaly can be determined if the cross-component dataare also available. To further illustrate the usefulness of theseconcepts, the foregoing analysis may also be used to detect an anomalyat a distance ahead of the drill bit. Analysis of Coaxial TransientResponses in Two-Layer Models FIG. 34 shows a coaxial tool withtransmitter-receiver spacing L placed in, for example, a vertical wellapproaching an adjacent bed that is the resistivity anomaly. The toolincludes both a transmitter coil T and a receiver coil R, which arewound around a common tool axis and are oriented in the tool axisdirection. The symbols σ₁ and σ₂ may represent the conductivities of twoformation layers.

To show that the transient EM method can be used as a look-aheadresistivity logging method, the transient response of the tool in atwo-layer earth model may be examined. There are three parameters thatmay be determined in the two-layer model. These are:

(1) the conductivity or resistivity (in the present example σ₁=0.1 S/mor R₁=10 ohm-m) of the local layer where the tool is placed;

(2) the conductivity or resistivity (in the present example σ₂=1 S/m orR₂=1 ohm-m) of an adjacent bed; and

(3) the distance of the tool to the layer boundary, for which in thepresent example d=1, 5, 10, 25, and 50 m are taken.

Under a more general circumstance, the relative direction of a boreholeand tool to the bed interface is not known.

The voltage response of the L=1 m (transmitter-receiver offset) coaxialtool at different distances (d) as a function of t is shown in FIG. 35.Though the difference is observed among responses at differentdistances, it is not straightforward to identify the resistivity anomalyfrom these responses.

The same voltage data of FIG. 35 is plotted in terms of the apparentconductivity (σ_(app)(t)) in FIG. 36. From this Figure, it is clear thatthe coaxial response can identify an adjacent bed of higher conductivityat a distance. Even a L=1 m tool can detect the bed at 10, 25-and 50-maway if low voltage response can be measured for 0.1-1 second long.

The σ_(app)(t) plot exhibits at least three parameters very distinctlyin the figure: the early time conductivity; the later time conductivity;and the transition time that moves as the distance (d) changes. In FIG.36, it should be noted that, at early time, the tool reads the apparentconductivity of 0.1 S/m that is of the layer just around the tool. Atlater time, the tool reads close to 0.55 S/m, an arithmetic averagebetween the conductivities of the two layers. The change of distance (d)is reflected in the transition time.

FIG. 37 illustrates the σapp(t) plot of the coaxial transient responsein the two-layer model of FIG. 34 for an L=1 m tool at differentdistances (d), except that the conductivity of the local layer (σ1) is 1S/m (R1=1 ohm-m) and the conductivity of the target layer (σ2) is 0.1S/m (R2=10 ohm-m). Again, the tool reads at early time the apparentconductivity of 1.0 S/m that is of the layer just around the tool. At alater time, the tool reads about 0.55 S/m, the same average conductivityvalue as in FIG. 36. The change of distance (d) is reflected in thetransition time.

Early time Conductivity (σ_(app)(t→0))

It is obvious that the early time conductivity is the conductivity ofthe local layer where the tool is located. At such an early time, thesignal reaches the receiver directly from the transmitter withoutinterfering with the bed boundary. Hence, it is affected only by theconductivity around the tool. Conversely, the layer conductivity can bemeasured easily by the apparent conductivity at an earlier time.

Late Time Conductivity (σ_(app)(t→∞))

On the other hand, the late time conductivity must be some average ofconductivities of both layers. At later time, nearly half of the signalscome from the formation below the tool and the other half from above thetool, if the time to travel the distance (d) of the tool to the bedboundary is small.

FIG. 38 compares the σapp(t) plot of FIG. 36 and FIG. 37 for L=01 m andd=01 m. The late time conductivity is determined solely by theconductivities of the two layers (al and σ2) alone. It is not affectedby where the tool is located in the two layers. However, because of thedeep depth of investigation, the late time conductivity is not readilyreached even at t=1 second, as shown in Table 31 for the same tool. Inpractice, the late time conductivity may have to be approximated byσapp(t=1 second) which slightly depends on d as illustrated in FIG. 39.

FIG. 40 compares the σ_(app)(t) plots for d=1 m but with differentspacing L. The σ_(app)(t) reaches a nearly constant late timeconductivity at later times as L increases. The late time conductivity(σ_(app)(t→∞) is nearly independent of L. However, the late timeconductivity defined at t=1 second, depends on the distance (d) as shownin FIG. 41

FIG. 42 compares the σ_(app)(t) plots for d=5 m and L=01 m but fordifferent resistivity ratios. This Figure shows that the late timeapparent conductivity is proportional to a, for the same ratio (σ₁/σ₂).For instance: $\begin{matrix}{{\sigma_{app}\left( {t->\infty} \right)}\left( {{{R_{1}\text{/}R_{2}} = 10};{R_{1} = {{10\quad{ohm}\text{-}m} = {2^{*}{\sigma_{app}\left( {t->\infty} \right)}\left( {{{R_{1}\text{/}R_{2}} = 10};{R_{1} = {20\quad{ohm}\text{-}m}}} \right)}}}} \right.} & (41)\end{matrix}$

FIG. 43 shows examples of the app(t) plots for d=5 m and L=01 m but fordifferent resistivity ratios while the target resistivity is fixed atR₂=1 ohm-m. The late time apparent conductivity at t=1 second isdetermined by the local layer conductivity as shown in FIG. 44.Numerically, the late time conductivity may be approximated by thearithmetic average of two-layer conductivities as:σ_(app)(t→∞;σ₁,σ₂)=σ₁+σ₂/2. This is reasonable considering that, withthe coaxial tool, the axial transmitter induces the eddy currentparallel to the bed boundary. At later time, the axial receiver receiveshorizontal current nearly equally from both layers. As a result, thelate time conductivity must see conductivity of both formations withnearly equal weight.

To summarize, the late time conductivity (σ_(app)(t→∞)) at t=1 secondcan be used to estimate the conductivity of the adjacent layer (σ₂) whenthe local conductivity near the tool (σ₁) is known, for instance, fromthe early time conductivity (σ_(app)(t→0)=σ₁). This is illustrated inFIG. 45.

Estimation of the Distance (d) to the Adjacent Bed

The transition time (t_(c)) at which the apparent conductivity startsdeviating from the local conductivity (σ₁) toward the late timeconductivity clearly depends on d, the distance of the tool to the bedboundary, as shown in FIG. 36 for a L=01 m tool.

For convenience, the transition time (t_(c)) is defined by the time atwhich the σ_(app)(t_(c)) takes the cutoff conductivity (σ_(c)), that is,in this example, the arithmetic average between the early time and thelate time conductivities: σ_(c)={σ_(app)(t→0)+σ_(app)(t→√))2. Thetransition time (t_(c)) is dictated by the ray-path (d) minus L/2 thatis, half the distance for the EM signal to travel from the transmitterto the bed boundary to the receiver, independently on the resistivity ofthe two layers. Conversely, the distance (d) can be estimated from thetransition time (t_(c)), as shown in FIG. 46 when L=01 m.

Image Presentation with the Apparent Conductivity

A plot of the apparent conductivity (σapp(z; t)) in both z- andt-coordinates may serve as an image presentation of the transient data,which represents apparent conductivity plots for the same tool atdifferent depths, as shown in FIG. 47. The z-coordinate represents thetool depth along the borehole. The σ_(app)(z; t) plot clearly helps tovisualize the approaching bed boundary as the tool moves along theborehole.

Analysis of Coplanar Transient Responses in Two-Layer Models

While the coaxial transient data were examined above, the coplanartransient data are equally useful as a look-ahead resistivity loggingmethod. FIG. 48 shows a coplanar tool with transmitter-receiver spacingL placed in a well approaching an adjacent bed that is the resistivityanomaly. On the coplanar tool, both a transmitter T and a receiver R areoriented perpendicularly to the tool axis and parallel to each other.The symbols σ₁ and σ₂ may represent the conductivities of two formationlayers.

Corresponding to FIG. 36 for coaxial tool responses where L=01 m, theapparent conductivity (σ_(app)(t)) for the coplanar responses is plottedin FIG. 49 for different tool distances from the bed boundary. It isclear that the coplanar response can also identify an adjacent bed ofhigher conductivity at a distance. Even a L=1 m tool can detect the bedat 10-, 25- and 50-m away if low voltage responses can be measured for0.1-1 second long. The σ_(app)(t) plot for the coplanar responsesexhibits three parameters equally as well as for the coaxial responses.Early time conductivity (σ_(app)(t→0))

It is also true for the coplanar responses that the early timeconductivity (σ_(app)(t→0)) is the conductivity of the local layer (σ₁)where the tool is located. Conversely, the layer conductivity can bemeasured easily by the apparent conductivity at earlier times.

Late Time Conductivity (σ_(app)(t→∞))

The late time conductivity (σ_(app)(t→∞)) is some average ofconductivities of both layers. The conclusions derived for the coaxialresponses apply equally well to the coplanar responses. However, thevalue of the late time conductivity for the coplanar responses is notthe same as for the coaxial responses. For coaxial responses, the latetime conductivity is close to the arithmetic average of two-layerconductivities in two-layer models. FIG. 49 shows the late timeconductivity (σ_(app)(t→∞)) for coplanar responses where d=05 m and L=01m but for different conductivities of the local layer while the targetconductivity is fixed at 1 S/m. Late time conductivity is determined bythe local layer conductivity, and is numerically close to the squareroot average as,$\sqrt{\sigma_{app}\left( {{{t->\infty};\sigma_{1}},\sigma_{2}} \right)} = {\frac{\sqrt{\sigma_{1}} + \sqrt{\sigma_{2}}}{2}.}$

To summarize, the late time conductivity (σapp(t→∞)) can be used toestimate the conductivity of the adjacent layer (σ₂) when the localconductivity near the tool (al) is known, for instance, from the earlytime conductivity (σapp(t→0)=σ1). This is illustrated in FIG. 51.

Estimation of the Distance (d) to the Adjacent Bed

The transition time at which the apparent conductivity starts deviatingfrom the local conductivity (σ₁) toward the late time conductivityclearly depends on the distance (d) of the tool to the bed boundary, asshown in FIG. 48.

The transition time (t_(c)) may be defined by the time at which theσ_(app)(t_(c)) takes the cutoff conductivity (σ_(c)) that is, in thisexample, the arithmetic average between the early time and the late timeconductivities: σ_(c)={σ_(app)(t→0)+σ_(app)(t→∞)}/2. The transition time(t_(c)) is dictated by the ray-path (d) minus L/2 that is, half thedistance for the EM signal to travel from the transmitter to the bedboundary to the receiver, independently of the resistivity of the twolayers.

Conversely, the distance (d) can be estimated from the transition time(t_(c)), as shown in FIG. 52 where L=1 m.

Fast Imaging Utilizing Apparent Conductivity

The use of the apparent conductivity and apparent dip may be used tocreate an “image” or representation of the formation features. This isaccomplished-by collecting transient apparent conductivity data atdifferent positions within the borehole. Utilizing distance anddirectional information as derived above, the collected data may be usedto create an image of the formation relative to the tool.

The first instance was to confirm the change in voltage response basedon the distance to the target formation using a coaxial toolinvestigating a two layer model in which the formation was parallel tothe axis of the tool. FIG. 53 depicts a coaxial tool in a two layerformation wherein the tool axis is parallel to the layer interface. Thiscan be a case of a horizontal well placement above the oil-water contactwhen the layers are horizontal and the tool is in the horizontalposition. The near layer is modeled as a brine saturated layer,resulting in a low resistivity (1 ohm-m). Alternatively, the model maybe used to represent the tool in a vertical well bore where the tool isbeing used for purposes of salt dome profiling and the salt dome isrepresented by a high resistivity layer located radially or sidewaysfrom the borehole. In this model, borehole effects are ignored due tothe large scale measurements being made. The tool of FIG. 53 is modeledas having two spacings, a 1 m and a 10 m spacing between transmitter andreceiver coils. Three basic parameters are used to characterize themodel: the conductivity of the near formation in which the tool islocated, σ₁, the resistivity of the target or distant formation, σ₂, andthe distance D to the interface with the formation of high resistivity.FIG. 54 is the voltage response in the two-layer model with an antennaspacing of 1 m, approximating the tool where, σ₁=1 S/m (R₁=1 ohm-m),σ₂=0.01 S/m (R₁=100 ohm-m). The change in the voltage response as afunction of distance can be clearly seen in FIG. 54.

Based on the voltage responses it is clear that the transient responsewould vary with the distance of the more resistive layer.

The next model utilized conductive near layer, a very resistive layer,and a further conductive layer. The configuration utilized is depictedin FIG. 55. Apparent resistivity (an inverse of apparent conductivity)from a coaxial tool at 10 m away from the resistive bed (salt) is shownin FIG. 55 for different salt bed thickness. The tool is modeled asbeing parallel to the interface with the resistive layer at a distanceof 10 m. The resistive bed thickness is varied from a fraction of ameter to 100 meters in thickness. The modeled apparent resistivityresponse is depicted in FIG. 56.

The first climb of R_(app)(t) is the response to the salt and takesplace at 10⁻⁴ s with L=1 m tool when the salt is at D=10 m away. If thesalt is fully resolved (by infinitely thick salt beyond D=10 m), theapparent resistivity should read 3 ohm-m asymptotically. The subsequentdecline of Rapp(t) is the response to a conductive formation behind thesalt (resistive bed). R_(app)(late t) is a function of conductive bedresistivity and salt thickness. If the time measurement is limited to10⁻² S, the decline of R_(app)(t) may not be detected for the saltthicker than 500 m.

With respect to the resistive bed resolution, the coaxial responds to athin (1-2m thick) bed. The time at which R_(app)(t) peaks or beginsdeclining depends on the distance to the conductive bed behind the salt.As noted previously, when plotted in terms of apparent conductivityσ_(app)(t), the transition time may be used to determine the distance tothe boundary beds.

A three-layer formation was also modeled. In this instance, theintermediate layer was a more conductive layer. The depiction of themodel is shown in FIG. 57. Therein the coaxial tool, having a 1 mspacing is located in a borehole in a formation having a resistivity of10 ohm-m and is located 10 m from a less resistive (more conductive)layer, having a resistivity of 1 ohm-m. A third layer is beyond theconductive bed and likewise has a resistivity of 10 ohm-m. Theconductive bed was modeled for varying thickness from fractions of ameter up to an infinite thickness. The conductive bed could consideredrepresentative of a shale layer. The apparent resistivity is set forthin FIG. 58.

The decrease in R_(app)(t) is due to the introduction of the shale(conductive) layer and appears as t→10⁻⁵ s. The shale response is fullyresolved by an infinitely thick conductive layer that approaches 3ohm-m. The subsequent rise in R_(app)(t) is in response to the resistiveformation beyond the shale layer. The transition time is utilized todetermine the distance to the interface between the second and thirdlayers. R_(app)(late t) is a function of conductive bed resistivity. Asthe conductive bed thickness increases, the time measurement mustlikewise be increased (>10⁻² s) in order to measure the rise ofR_(app)(t) for conductive layers thicker than 100 m.

Another three layer model is set forth in FIG. 59, wherein the coaxialtool is in a conductive layer (1 ohm-m), and a highly resistive layer(100 ohm-m) as might be found in a salt dome. The two layers areseparated by a layer having an intermediate resistance (10 ohm-m) ofvarying thickness. The apparent resistivity response is depicted in FIG.60.

The response to the intermediate resistive layer is seen at 10⁻⁴ S,where R_(app)(t) increases. If the intermediate layer is fully resolvedby an infinitely thick bed, the apparent resistivity approaches a 2.6ohm-m asymptote. As noted in FIG. 60, the R_(app)(t) undergoes a secondstage increase in response to the 100 ohm-m highly resistive layer.Based on the transition time, the distance to the interface isdetermined to be 110 m.

Though complex, the apparent resistivity or apparent conductivity in theabove examples delineates the presence of multiple layers. When theapparent resistivity plots (t, R_(app)(t))at different tool positionsare arranged together, the whole plot may be used as an image log toview the formation geometry, even if the layer resistivity may not beimmediately accurately determined. An example is shown in FIG. 61wherein a 3 layer model is used in conjunction with a coaxial toolhaving a 1 m spacing is in two differing positions in the formation. Theresults are plotted on FIG. 62.

The apparent resistivity R_(app)(t) is plotted at various points as thecoaxial tool approaches the resistive layer. In starting in the 10 ohm-mlayer, the drop in R_(app)(t) is attributable to the 1 ohm-m layer andthe subsequent increase in R_(app)(t) is attributable to the 100 ohm-mlayer. Curves may readily be fitted to the inflection points to identifythe responses to the various beds, effectively imaging the formation.Moreover, the 1 ohm-m curve may be readily attributable to direct signalpick up between the transmitter and receiver when the tool is located inthe 1 ohm-m bed.

In yet another example, the apparent. dip θ_(app)(t) may be used togenerate an image log. In the right hand side of FIG. 63 a coaxial toolis seen as approaching a highly resistive formation at a dip angle ofapproximately 30 degrees. The apparent dip response is shown at the lefthand side of FIG. 63. As noted previously, the time at which theapparent dip response occurs is indicative of the distance to theformation. When the responses for different distances are plottedtogether, a curve may be drawn indicative of the response as the toolapproaches the bed, as shown in the left hand side of FIG. 63.

Thus an image of the formation may be created using apparentconductivity/resistivity and dip without the additional processingrequired for inversion and extraction of information. This informationis capable of providing geosteering queues as well as the ability toprofile subterranean formations.

The present invention has been described in relation to particularembodiments, which are intended in all respects to be illustrativerather than restrictive. Alternative embodiments will become apparent tothose skilled in the art to which the present invention pertains withoutdeparting from its scope.

From the foregoing, it will be seen that this invention is one welladapted to attain all the ends and objects set forth above, togetherwith other advantages, which are obvious and inherent to the system andmethod. It will be understood that certain features and sub-combinationsare of utility and may be employed without reference to other featuresand sub-combinations. This is contemplated and within the scope of theclaims.

1. A method for imaging a subterranean formation traversed by a wellboreusing a tool comprising a transmitter for transmitting electromagneticsignals through the formation and a receiver for detecting responsesignals, the methods comprising steps: (a) bringing the tool to a firstposition inside the wellbore; (b) energizing the transmitter topropagate an electromagnetic signal into the formation; (c) detecting aresponse signal that has propagated through the formation; (d)calculating a derived quantity for the formation based on the detectedresponse signal for the formation; (e) plotting the derived quantity forthe formation against time; (f) moving the tool to at least one otherposition within the wellbore and repeating steps (b) to (f) followed bystep (g) or (i); (g) creating an image of the formation within thesubterranean formation based on plots resulting from step (f).
 2. Themethod of claim 1, wherein step (f) includes plotting the derivedquantity for the formation calculated in respect of the tool at the atleast one other position in the same plot as the plot wherein thederived quantity for the formation calculated in respect of the tool atthe first position was plotted, and wherein creating the image of theformation features of step (i) includes identifying one or moreinflection points on each plotted derived quantity and fitting a curveto the one or more inflection points.
 3. The method of claim 1, whereinstep (i) is followed by step (j) moving the tool to at least one moreother position within the wellbore and repeating steps (b) to (f)followed by step (g) or (i).
 4. The method of claim 1, wherein afterenergizing a transient change is induced in the transmittedelectromagnetic signal, and wherein plotting the derived quantity forthe formation against time comprises plotting the derived quantity forthe formation against time lapsed after the transient change.
 5. Themethod of claim 4, wherein the transient change induced in thetransmitted electromagnetic signal comprises terminating the signal. 6.The method of claim 1, wherein the derived quantity is one of apparentconductivity and apparent resistivity and apparent dip angle andapparent azimuth angle.
 7. The method of claim 1, wherein the image ofthe formation represents one or more formation layers each displaying amutually different formation feature.
 8. The method of claim 7, whereinfor each position of the tool in the wellbore a distance is determinedfrom the tool to at least one formation layer.
 9. The method of claim 7,wherein determining the distance comprises determining a time in whichthe one of apparent conductivity and apparent resistivity begins todeviate from the corresponding one of apparent conductivity and apparentresistivity of formation in which the device is located.
 10. The methodof claim 1, wherein detecting a response signal comprises detecting aninduced voltage response signal.
 11. The method of claim 10, whereincalculating the derived quantity includes converting the induced voltagesignal to the derived quantity.
 12. A method for determining at leastfirst and second distances from a device to at least a first layer and asecond layer in a formation, whereby at least one of the first andsecond layers comprises a resistivity or conductivity anomaly, thedevice comprising a transmitter for transmitting electromagnetic signalsthrough the formation and a receiver for detecting responses, the methodcomprising: bringing the device inside a wellbore in the formation;transmitting an electromagnetic signal using the transmitter;calculating one of apparent conductivity and apparent resistivity basedon a receiver-detected response; monitoring the one of apparentconductivity and apparent resistivity over time; and determining thefirst and second distances to the anomaly based on observed changes ofthe one of apparent conductivity and apparent resistivity.
 13. Themethod of claim 12, wherein the observed changes each include adeflection point.
 14. The method of claim 12, wherein transmitting theelectromagnetic signal includes inducing a transient change in thetransmitted electromagnetic signal and wherein monitoring the one ofapparent conductivity and apparent resistitivity over time comprisesmonitoring the apparent conductivity during a time range following thetransient change.
 15. The method of claim 14, wherein the transientchange comprises terminating the transmission of the electromagneticsignal, and wherein the receiver-detected response comprises atime-resolved decay of a signal during the time range following thetransient change.
 16. The method of claim 14, wherein the time range isselected based on a separation distance between the transmitter and thereceiver.
 17. The method of claim 14, wherein the time range runs fromat least 10-8 seconds to at least 0.1 seconds, preferably to at least 1seconds.
 18. The method of claim 14, wherein the time range is selectedsufficiently long to have the one of apparent conductivity and apparentresistivity reach an asymptotic value, whereby the distance to theanomaly is determined when the one of apparent conductivity and apparentresistivity reaches an asymptotic value.
 19. The method of claim 12,wherein the receiver-detected response comprises a voltage response. 20.The method of claim 12, wherein calculating the one of apparentconductivity and apparent resistivity includes converting an inducedvoltage signal to an average conductivity of the formation.
 21. Themethod of claim 12, wherein calculating the one of apparent conductivityand apparent resistivity comprises evaluating one of a correspondingapparent conductivity and apparent resistivity of the formation in whichthe device is located.
 22. The method of claim 12, wherein calculatingthe one of apparent conductivity and apparent resistivity comprisesevaluating the corresponding one of apparent conductivity and apparentresistivity of the formation ahead of the device.
 23. The method ofclaim 12, wherein determining the first distance to the anomalycomprises determining a time in which the one of apparent conductivityand apparent resistivity begins to deviate from the corresponding one ofapparent conductivity and apparent resistivity of the formation in whichthe device is located.
 24. The method of claim 12, wherein theelectromagnetic signal is transmitted in the direction of the anomaly.25. The method of claim 12, wherein the device comprises a logging tool.26. The method of claim 12, wherein the device is provided in ameasurement-while-drilling or logging-while-drilling section of a drillstring located uphole relative to a drill bit.
 27. The method of claim12, wherein the first layer is formed by an anomaly in the formationwherein the device is positioned, whereby the, second layer is comprisedof a similar formation as that wherein the device is positioned.
 28. Themethod of claim 12, wherein the difference between the first and seconddistances corresponds to the thickness of the first layer.
 29. Themethod of claim 12, wherein the monitored one of apparent conductivityand apparent resistivity is plotted against time and device issubsequently moved to another position in the wellbore, after which anelectromagnetic signal is again transmitted using the transmitter; oneof apparent conductivity and apparent resistivity based on areceiver-detected response is again calculated; the one of apparentconductivity and apparent resistivity is again monitored over time; andagain plotted against time; after which an image of the formation withinthe subterranean formation is created based on the plots.
 30. The methodof claim 29, wherein the again monitored one of apparent conductivityand apparent resistivity is plotted in the same plot as the plot whereinthe one of apparent conductivity and apparent resistivity was originallyplotted, and wherein creating the image of the formation includesidentifying two or more inflection points on each plotted calculated oneof apparent conductivity and apparent resistivity curve and fitting acurve to the two or more inflection points.